Dalitz studied in Ref. [1]} the potential scattering of an electron with a Yukawa potential \(V(r)={e_1 e_2e^{-\mu r}}/{4\pi r}\)  in the limit \(\mu \rightarrow 0\)  both in the nonrelativistic and relativistic cases, the former up to \({\cal O}(e^3)\)  and the latter up to \({\cal O}(e^2)\) . He then conjectured that the infrared divergences summed up giving rise to a phase factor.
The raise of this phase was demonstrated by Weinberg in Ref. [2]} for QED and gravity. In the limit of potential scattering the full extent of virtual infrared photons for any process \(\alpha \rightarrow \beta \)  is to give rise to a phase factor in the \(S\) -matrix given by


SS0(L)={12L A(q)} ,

where \(S_{\beta \alpha }^0({\cal L})\)  has no infrared divergences and \(A(q)\)  is given in Ref. [2]}. The scale \({\cal L}\)  is introduced in this reference to separate between soft and hard physics. The integral in Eq. () is worked out in Ref. [2]}, such that each different pair of particles \(n\)  and \(m\)  in the initial or final state contributes to the \(S\)  matrix with a diverging phase factor, which has then to be summed over all these pairs. The neat result is


SS0(L)=Sc={i2pairsenemnmL} ,

with \(\beta _{nm}=\frac{\sqrt{(p_n\cdot p_m)^2-(m_n m_m)^2}}{p_n\cdot p_m}\)  , the invariant relative velocity between the particles \(n\)  and \(m\)  of momenta \(p_n\)  and \(p_m\) , respectively. From this equation it is clear that the scale \({\cal L}\)  must be proportional to \(2p\)  since the resummed exchange of soft photons of mass \(\mu \)  (which acts as an infrared regulator) implies a left-hand cut (LC) starting at \(4p^2=-\mu ^2\) . We then write


L=2pa ,

with the constant \(a\)  being necessarily independent of \(p\) , and only enters logarithmically with \(\ln a={\cal O}(1)\)  [5]}, [6]}. For graviton-graviton scattering the analogous to Eq. () is


Sc {-ipairsG mn mm (1+nm2) nm[1-nm2]1/2L} .
